Chapter 7: Infinite abelian groups For infinite abelian
... 2. If A and B are subsets of I, A is a subset of B, and A is an element of F , then B is also an element of F . 3. If A and B are elements of I, then so is the intersection of A and B. F is an ultrafilter if additionally 4. If A is a subset of I, then either A or I − A is an element of F . Propertie ...
... 2. If A and B are subsets of I, A is a subset of B, and A is an element of F , then B is also an element of F . 3. If A and B are elements of I, then so is the intersection of A and B. F is an ultrafilter if additionally 4. If A is a subset of I, then either A or I − A is an element of F . Propertie ...
1. Introduction 1 2. Simplicial and Singular Intersection Homology 2
... Proof. We will prove the theorem for ordinary cohomology. For Borel-Moore cohomology, we can either use Poincare duality or use the fact that there is a natural chain map I p,BM Si (C(X)) → I p Si (C(X))/I p Si (C(X)\{v}) which induces an isomorphism on homology and subsequently use the long exact s ...
... Proof. We will prove the theorem for ordinary cohomology. For Borel-Moore cohomology, we can either use Poincare duality or use the fact that there is a natural chain map I p,BM Si (C(X)) → I p Si (C(X))/I p Si (C(X)\{v}) which induces an isomorphism on homology and subsequently use the long exact s ...
A Crash Course in Topological Groups
... The underlying additive group of a Banach space (or more generally, topological vector space): Rn , Lp (X , µ), C0 , Cc (X ), ... Groups of homeomorphisms of a ‘nice’ topological space, or diffeomorphisms of a smooth manifold, can be made into topological groups. Any group taken with the discrete to ...
... The underlying additive group of a Banach space (or more generally, topological vector space): Rn , Lp (X , µ), C0 , Cc (X ), ... Groups of homeomorphisms of a ‘nice’ topological space, or diffeomorphisms of a smooth manifold, can be made into topological groups. Any group taken with the discrete to ...
Topological Groups Part III, Spring 2008
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
... group for each α ∈ A. Their complete direct product is G = α Gα equipped with the usual product topology τG and with multiplication given by (x ×G y)α = xα ×α yα . Definition 4.2. Suppose A is non-empty and (Gα , ×α ) is a group Q for each α ∈ A. Their direct product is the subgroup of the group G = ...
The fundamental groupoid as a topological
... A:n(X, x)-+G{x}. Corollary 6. Let X be path-connected and x e X. The following conditions are equivalent. (i) StGx is path-connected. (ii) A: n(X, x)-+G{x} is surjective. (iii) 5* : n(G, lx)^>n(X, x) is surjective. Proof. These follow immediately from the exact homotopy sequences of the fibrations d ...
... A:n(X, x)-+G{x}. Corollary 6. Let X be path-connected and x e X. The following conditions are equivalent. (i) StGx is path-connected. (ii) A: n(X, x)-+G{x} is surjective. (iii) 5* : n(G, lx)^>n(X, x) is surjective. Proof. These follow immediately from the exact homotopy sequences of the fibrations d ...
Sheaves of Modules
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d59dda8, compiled on Apr 24, ...
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d59dda8, compiled on Apr 24, ...
Sheaves of Modules
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d9096d4, compiled on Oct 19, ...
... In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Zmodules. Basic references are [Ser55], [DG67] and [AGV71]. This is a chapter of the Stacks Project, version d9096d4, compiled on Oct 19, ...
Chapter 7 Duality
... in SmS , admits a duality involution. In particular, if S = Spec(k), and if one has resolution of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-( ...
... in SmS , admits a duality involution. In particular, if S = Spec(k), and if one has resolution of singularities for k-varieties, then the category DM(k) has a duality involution, making DM(k) a rigid triangulated tensor category. We then give some applications of the duality involution: in (7.4.4)-( ...
Basic Modern Algebraic Geometry
... objects be the set of all open subsets of X, and for two open subsets U and V we let Hom(U, V ) be the set whose only element is the inclusion mapping if U ⊆ V , and ∅ otherwise. This is a category, as is easily verified. If U ⊆ X is an open subset, then the category Top(X)U is nothing but the categ ...
... objects be the set of all open subsets of X, and for two open subsets U and V we let Hom(U, V ) be the set whose only element is the inclusion mapping if U ⊆ V , and ∅ otherwise. This is a category, as is easily verified. If U ⊆ X is an open subset, then the category Top(X)U is nothing but the categ ...
Equivariant K-theory
... As a group R(G) is the free abelian group generated by the set G of simple G-modules. In general K^(X) is an algebra over R(G), because any G-space X has a natural map on to a point. (The morphism R(G) -^K^(X) is just Ml-^M.) (ii) Ko(G/H) ^ R(H) when H is a closed subgroup ofG. For we have seen that ...
... As a group R(G) is the free abelian group generated by the set G of simple G-modules. In general K^(X) is an algebra over R(G), because any G-space X has a natural map on to a point. (The morphism R(G) -^K^(X) is just Ml-^M.) (ii) Ko(G/H) ^ R(H) when H is a closed subgroup ofG. For we have seen that ...
A SHEAF THEORETIC APPROACH TO MEASURE THEORY Matthew Jackson by
... The study of category theory allows for the development of techniques that can apply simultaneously in all of these settings. Categories have been studied extensively, for example in Mac Lane [18], Barr and Wells [1, 2], and McLarty [20]. Definition 1. A category C consists of a collection OC of ob ...
... The study of category theory allows for the development of techniques that can apply simultaneously in all of these settings. Categories have been studied extensively, for example in Mac Lane [18], Barr and Wells [1, 2], and McLarty [20]. Definition 1. A category C consists of a collection OC of ob ...
equivariant homotopy and cohomology theory
... Chapter IV gives two proofs of the rst main theorem of equivariant algebraic topology, which goes under the name of \Smith theory": any xed point space of an action of a nite p-group on a mod p homology sphere is again a mod p homology sphere. One proof uses ordinary (or Bredon) equivariant cohom ...
... Chapter IV gives two proofs of the rst main theorem of equivariant algebraic topology, which goes under the name of \Smith theory": any xed point space of an action of a nite p-group on a mod p homology sphere is again a mod p homology sphere. One proof uses ordinary (or Bredon) equivariant cohom ...
Vector bundles and torsion free sheaves on degenerations of elliptic
... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...
... In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projectiv ...
slides - Math User Home Pages
... i) Let F : Sets → Ab be a functor assigning a to a set X the free group generated on X. It induces a functor F∗ : sSets → sAb Σ 7→ F ◦ Σ Intuitively, one can think of F∗ (Σ)n as of the free abelian group on the n-simplicies of the ”simplcial complex“ Σ. ii) Let A be a simplicial abelian group. We ca ...
... i) Let F : Sets → Ab be a functor assigning a to a set X the free group generated on X. It induces a functor F∗ : sSets → sAb Σ 7→ F ◦ Σ Intuitively, one can think of F∗ (Σ)n as of the free abelian group on the n-simplicies of the ”simplcial complex“ Σ. ii) Let A be a simplicial abelian group. We ca ...
NOTES ON NON-ARCHIMEDEAN TOPOLOGICAL GROUPS
... we define u1 u2 = (a1 + a2 + f1 (x2 ), x1 + x2 , f1 + f2 ) where, f1 (x2 ) = w(x2 , f1 ). Then H(w) becomes a Hausdorff topological group which is said to be a generalized Heisenberg group (induced by w). Let G be a topological group and let w : E × F → A be a continuous biadditive mapping. A contin ...
... we define u1 u2 = (a1 + a2 + f1 (x2 ), x1 + x2 , f1 + f2 ) where, f1 (x2 ) = w(x2 , f1 ). Then H(w) becomes a Hausdorff topological group which is said to be a generalized Heisenberg group (induced by w). Let G be a topological group and let w : E × F → A be a continuous biadditive mapping. A contin ...
Scheme representation for first-order logic
... from this representation. The most familiar example of such a representation is Grothendieck’s theorem that every commutative ring R is isomorphic to the ring of global sections of a certain sheaf over the Zariski topology on R. Together with a locality condition, this is essentially the constructi ...
... from this representation. The most familiar example of such a representation is Grothendieck’s theorem that every commutative ring R is isomorphic to the ring of global sections of a certain sheaf over the Zariski topology on R. Together with a locality condition, this is essentially the constructi ...
EBERLEIN–ŠMULYAN THEOREM FOR ABELIAN TOPOLOGICAL
... In many branches of mathematics the notion of compactness appears as a fundamental tool, and the optimal situation is when it can be used in its sequential version. Unfortunately this is not always the case, and there is a strong need to look for classes of topological spaces where compactness is eq ...
... In many branches of mathematics the notion of compactness appears as a fundamental tool, and the optimal situation is when it can be used in its sequential version. Unfortunately this is not always the case, and there is a strong need to look for classes of topological spaces where compactness is eq ...
STABLE HOMOTOPY THEORY 1. Spectra and the stable homotopy
... Definition 1.30. The homology theory associated to an object E ∈ Ob SH is the functor SH → Abgr defined by En (Z) := [S[n], Z ∧ E]SH . Example 1.31. Taking S = Σ∞ S 0 gives a homology theory Z 7→ [S[n], Z]; this should coincide with the stable homotopy functor. The restriction of the homology theory ...
... Definition 1.30. The homology theory associated to an object E ∈ Ob SH is the functor SH → Abgr defined by En (Z) := [S[n], Z ∧ E]SH . Example 1.31. Taking S = Σ∞ S 0 gives a homology theory Z 7→ [S[n], Z]; this should coincide with the stable homotopy functor. The restriction of the homology theory ...
12. Fibre products of schemes We start with some basic properties of
... Definition 12.14. Let X be a scheme and U an open subset of X. Then the pair (U, OU = OX |U ) is a scheme, which is called an open subscheme of X. An open immersion is a morphism f : X −→ Y which induces an isomorphism of X with an open subset of Y . Definition 12.15. A closed immersion is a morphi ...
... Definition 12.14. Let X be a scheme and U an open subset of X. Then the pair (U, OU = OX |U ) is a scheme, which is called an open subscheme of X. An open immersion is a morphism f : X −→ Y which induces an isomorphism of X with an open subset of Y . Definition 12.15. A closed immersion is a morphi ...
Metrizability of hereditarily normal compact like groups1
... is called unseparable in X if for every pair of open sets U, V of X such that A ✓ U and B ✓ V , one has U \ V 6= ;. ...
... is called unseparable in X if for every pair of open sets U, V of X such that A ✓ U and B ✓ V , one has U \ V 6= ;. ...
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... sets of Spec(R) in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf Spec(R) with the structure sheaf of the variety V , showing that the two objects are identical in every respect except for the minor detail of Spec(R) having more points ...
... sets of Spec(R) in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf Spec(R) with the structure sheaf of the variety V , showing that the two objects are identical in every respect except for the minor detail of Spec(R) having more points ...
Proper actions on topological groups: Applications to quotient spaces
... connected components endowed with the quotient topology is compact. Corollary 1.6. Let P be a topological property stable under open perfect maps and also inherited by closed subsets. Assume that X is a paracompact group with the property P and let G be an almost connected subgroup of X. Then the qu ...
... connected components endowed with the quotient topology is compact. Corollary 1.6. Let P be a topological property stable under open perfect maps and also inherited by closed subsets. Assume that X is a paracompact group with the property P and let G be an almost connected subgroup of X. Then the qu ...
Compact groups and products of the unit interval
... It is proved that if G is a compact connected Hausdorff group of uncountable weight, w(G), then G contains a homeomorphic copy of [0,1]W{O). From this it is deduced that such a group, G, contains a homeomorphic copy of every compact Hausdorff group with weight w(G) or less. It is also deduced that e ...
... It is proved that if G is a compact connected Hausdorff group of uncountable weight, w(G), then G contains a homeomorphic copy of [0,1]W{O). From this it is deduced that such a group, G, contains a homeomorphic copy of every compact Hausdorff group with weight w(G) or less. It is also deduced that e ...
On the category of topological topologies
... bitrary union continuous maps (we use the symbol Y" for the set O Y topologized by T ). Isbell [8, 12] has proved that any pair of adjoint endofunctors F j G of Top is determined (up to functorial isomorphism) by a pair ( Y, Y*,), where Y = F ( { * }) and Y*=G(2),with 2 aSierpinski furthermore G(2) ...
... bitrary union continuous maps (we use the symbol Y" for the set O Y topologized by T ). Isbell [8, 12] has proved that any pair of adjoint endofunctors F j G of Top is determined (up to functorial isomorphism) by a pair ( Y, Y*,), where Y = F ( { * }) and Y*=G(2),with 2 aSierpinski furthermore G(2) ...
The Group of Extensions of a Topological Local Group
... E 0 = {(e, x0 )|π(e) = γ(x0 ), e ∈ E, x0 ∈ X 0 }; E 0 is a sublocal group of E ⊕ X 0 . By [5, Proposition 2.22], E 0 is a topological local group. We define π 0 : E 0 → X 0 , π 0 (e, x0 ) = x0 , σ : E 0 → E, σ(e, x0 ) = e, η 0 : C → kerπ ⊕ {1X 0 }, η 0 (n) = (η(n), 1X 0 ). Since π is onto then so is ...
... E 0 = {(e, x0 )|π(e) = γ(x0 ), e ∈ E, x0 ∈ X 0 }; E 0 is a sublocal group of E ⊕ X 0 . By [5, Proposition 2.22], E 0 is a topological local group. We define π 0 : E 0 → X 0 , π 0 (e, x0 ) = x0 , σ : E 0 → E, σ(e, x0 ) = e, η 0 : C → kerπ ⊕ {1X 0 }, η 0 (n) = (η(n), 1X 0 ). Since π is onto then so is ...